Cognitive and metacognitive activity in mathematical problem solving: prefrontal and parietal patterns

Students were taught an algorithm for solving a new class of mathematical problems. Occasionally in the sequence of problems, they encountered exception problems that required that they extend the algorithm. Regular and exception problems were associated with different patterns of brain activation. Some regions showed a Cognitive pattern of being active only until the problem was solved and no difference between regular or exception problems. Other regions showed a Metacognitive pattern of greater activity for exception problems and activity that extended into the post-solution period, particularly when an error was made. The Cognitive regions included some of parietal and prefrontal regions associated with the triple-code theory of (Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506) and associated with algebra equation solving in the ACT-R theory (Anderson, J. R. (2005). Human symbol manipulation within an 911 integrated cognitive architecture. Cognitive science, 29, 313–342. Metacognitive regions included the superior prefrontal gyrus, the angular gyrus of the triple-code theory, and frontopolar regions.     

Dynamics of metacognitive judgments: pre- and postretrieval mechanisms

Two experiments examined the temporal course of pre- and postretrieval mechanisms using a new kind of dynamic metacognitive judgment. Experiment 1 presented participants with primed and unprimed triples of remote associates to a target word and required them to provide repeated metacognitive judgments, 4 times during a 12-s interval, about the likelihood that they would later recognize the target. Both familiarity with the words and the processing time were associated with changes in metacognitive evaluations. Experiment 2 placed pre- and postretrieval mechanisms in opposition by transforming an element of a previously primed question. For transformed questions this led to high initial ratings, which decreased over time, while for novel questions the ratings were initially lower and increased with time. The results are discussed in terms of pre- and postretrieval mechanisms interacting over time.     

Insight versus analysis: Evidence for diverse methods in problem solving

Since the Gestalt psychologists made the distinction approximately 100 years ago, psychologists have differentiated between solving problems through analysis versus insight. The present paper presents evidence to support the idea that, rather than conceptualising insight versus analysis as distinct modes of solving problems, it is more useful to conceive of insight and analysis as two approaches within a set of possible solving methods. In the present research, 60 participants solved insight problems while thinking aloud, which provided evidence concerning the processes underlying problem solution. Comparison with performance of a nonverbalisation control group (n = 35) indicated no negative effects of thinking aloud on insight in problem solving. The results supported the idea that various methods are utilised in solving insight problems. The “classic” impasse–restructuring–insight sequence occurred in only a small minority of solutions. A number of other solution methods were found, ranging from relatively direct applications of knowledge, to various heuristic methods, to restructuring arising from new information gleaned from a failed solution. It is concluded that there is not a sharp distinction between solving a problem through analysis versus insight, and implications of that conclusion are discussed.     

Characterizing the intuitive representation in problem solving: Evidence from evaluating mathematical strategies

Two experiments were conducted to investigate the nature of the intuitive problem representation used in evaluating mathematical strategies. The first experiment tested between two representations: a representation composed of principles and an integrated representation. Subjects judged the correctness of unseen math strategies based only on the answers they produced for a set of temperature mixture problems. The distance of the given answers from the correct answers and whether the answers violated one of the principles of temperature mixture were manipulated. The results supported the principle representation hypothesis. In the second experiment we manipulated subjects’ understanding of an acid mixture task with a brief paragraph of instruction on one of the principles. Subjects then completed an estimation task intended to measure their understanding of the problem domain. The evaluation task from the first experiment was then presented, but with acid mixture instead of temperature mixture. The results showed that intuitive understanding of the domain mediates the effect of instruction on evaluating problems. Additionally, the results supported the hypothesis that subjects perform a mapping process between their intuitive understanding and math strategies.     

Dynamics and constraints in insight problem solving

This article reports 2 experiments that investigated performance on a novel insight problem, the 8-coin problem. The authors hypothesized that participants would make certain initial moves (strategic moves) that seemed to make progress according to the problem instructions but that nonetheless would guarantee failure to solve the problem. Experiment 1 manipulated the starting state of the problem and showed that overall solution rates were lower when such strategic moves were available. Experiment 2 showed that failure to capitalize on visual hints about the correct first move was also associated with the availability of strategic moves. The results are interpreted in terms of an information-processing framework previously applied to the 9-dot problem. The authors argue that in addition to the operation of inappropriate constraints, a full account of insight problem solving must incorporate a dynamic that steers solution-seeking activity toward the constraints.     

Intuition in insight and noninsight problem solving

People’s metacognitions, both before and during problem solving, may be of importance in motivating and guiding problem-solving behavior. These metacognitions could also be diagnostic for distinguishing among different classes of problems, each perhaps controlled by different cognitive processes. In the present experiments, intuitions on classic insight problems were compared with those on noninsight and algebra problems. The findings were as follows: (1) subjective feeling of knowing predicted performance on algebra problems but not on insight problems; (2) subjects’ expectations of performance greatly exceeded their actual performance, especially on insight problems; (3) normative predictions provided a better estimate of individual performance than did subjects’ own predictions, especially on the insight problems; and, most importantly, (4) the patterns-of-warmth ratings, which reflect subjects’ feelings of approaching solution, differed for insight and noninsight problems. Algebra problems and noninsight problems showed a more incremental pattern over the course of solving than did insight problems. In general, then, the data indicated that noninsight problems were open to accurate predictions of performance, whereas insight problems were opaque to such predictions. Also, the phenomenology of insight-problem solution was characterized by a sudden, unforeseen flash of illumination. We propose that the difference in phenomenology accompanying insight and noninsight problem solving, as empirically demonstrated here, be used to define insight.     

Hindsight bias: How knowledge and heuristics affect our reconstruction of the past

Once people know the outcome of an event, they tend to overestimate what could have been anticipated in foresight. Although typically considered to be a robust phenomenon, this hindsight bias is subject to moderating circumstances. In their meta-analysis, Christensen-Szalanski and Willham (1991) observed that the more experience people have with the task under consideration, the smaller is the resulting hindsight bias. This observation is one benchmark against which the explanatory power of process models of hindsight bias can be measured. Therefore, we used it to put the recently proposed RAFT model (Hoffrage, Hertwig, & Gigerenzer, 2000) to another test. Our findings were consistent with the "expertise effect." Specifically, we observed--using computer simulations of the RAFT model--that the more comprehensive people's knowledge is in foresight, the smaller is their hindsight bias. In addition, we made two counterintuitive observations: First, the relation between foresight knowledge and hindsight bias appears to be independent of how knowledge is processed. Second, even if foresight knowledge is false, it can reduce hindsight bias. We conclude with a discussion of the functional value of hindsight bias.     

Hindsight bias: how knowledge and heuristics affect our reconstruction of the past

Once people know the outcome of an event, they tend to overestimate what could have been anticipated in foresight. Although typically considered to be a robust phenomenon, this hindsight bias is subject to moderating circumstances. In their meta-analysis, Christensen-Szalanski and Willham (1991) observed that the more experience people have with the task under consideration, the smaller is the resulting hindsight bias. This observation is one benchmark against which the explanatory power of process models of hindsight bias can be measured. Therefore, we used it to put the recently proposed RAFT model (Hoffrage, Hertwig, & Gigerenzer, 2000) to another test. Our findings were consistent with the "expertise effect." Specifically, we observed-using computer simulations of the RAFT model-that the more comprehensive people's knowledge is in foresight, the smaller is their hindsight bias. In addition, we made two counterintuitive observations: First, the relation between foresight knowledge and hindsight bias appears to be independent of how knowledge is processed. Second, even if foresight knowledge is false, it can reduce hindsight bias. We conclude with a discussion of the functional value of hindsight bias.     

Priming insight in groups: Facilitating and inhibiting solving an ambiguously worded insight problem

We extend research on the priming of insight by studying group problem solving. Groups of 2–4 participants tried to solve an ambiguously worded problem in the presence of a prime that reinforced the dominant but incorrect interpretation of the problem, a prime that reinforced the uncommon but correct interpretation, or no prime. The paradigm involved participants asking questions of the experimenter that could only be answered “yes” or “no.” In Experiment 1, the prime was present throughout the solving period; in Experiment 2, it was removed prior to the solving period. In both experiments, the primes had their predicted effects. Patterns in the time taken to solve the problem supported the idea that groups stuck at the impasse were more or less able to restructure the problem, depending on the environmental context. Data from the questions asked and questionnaires converged with time taken to solve the problem, consistent with the view that restructuring a problem is an automatic process that produces insight. A comparison of the group data in Experiment 1 with individually tested participants’ data revealed that the insight of the groups benefited from their being able to recognize lines of questions to follow, to listen to answers to questions asked, and to evaluate and reject errors or assumptions.     

Influence of situational and conceptual rewording on word problem solving

Background. Studies on rewording word problems can be grouped into two main groups: situational rewording, in which the situation denoted by the text is described more richly, and conceptual rewording, in which the underlying semantic relations are highlighted. Aims. Our aims are to define and distinguish these two kinds of rewording and to test empirically their relative effectiveness in two different studies. Sample. In the first study, 79 third graders, 64 fourth graders and 65 fifth graders took part; the sample for Study 2 was similar. Method. In Study 1, children were asked to solve both easy and difficult two‐step change problems in three different versions: standard, situational and conceptual rewording. In Study 2, three different versions of the situational version were compared: one with only temporal elaborations, one with only causal elaborations and a ‘complete’ version combining both elaborations. Results. In Study 1, conceptually reworded problems elicited the best results, especially among younger children and for difficult two‐step problems. Neither in Study 1 nor in Study 2 did the situationally reworded problems yield better performance than standard items. Conclusion. Only conceptual rewording has proved to be useful for improving children's performance, especially among younger children and for difficult problems. The lack of impact of situational rewording cannot be explained in terms of the length of the resulting text.     

Swinging into thought: Directed movement guides insight in problem solving

Can directed actions unconsciously influence higher order cognitive processing? We investigated how movement interventions affected participants’ ability to solve a classic insight problem. The participants attempted to solve Maier’s two-string problem while occasionally taking exercise breaks during which they moved their arms either in a manner related to the problem’s solution (swing group) or in a manner inconsistent with the solution (stretch group). Although most of the participants were unaware of the relationship between their arm movement exercises and the problem-solving task, the participants who moved their arms in a manner that suggested the problem’s solution were more likely to solve the problem than were those who moved their arms in other ways. Consistent with embodied theories of cognition, these findings show that actions influence thought and, furthermore, that we can implicitly guide people toward insight by directing their actions.     

Effects of self‐explanation as a metacognitive strategy for solving mathematical word problems1

Abstract: This study examined how a metacognitive strategy known as self‐explanation influences word problem solving in elementary school children. Participants were 79 sixth‐graders. They were assigned to one of three groups: the self‐explanation group, the self‐learning group, or the control group. Students in each group performed a ratio word problem test and a transfer test. The results showed that students in the self‐explanation group outperformed students in the other two groups on both the ratio word problem test and on the transfer test. In addition, high explainers who generated more self‐explanations relating to deep understanding of worked‐out examples outperformed low explainers on both ratio word problem and transfer tests. The self‐explanation effect is discussed.     

Effects of meaningfulness and organization on problem solving and computability judgments

When subjects were required to calculate answers for computable problems and answer questions, an interaction was found corresponding to that obtained by Kieras and Greeno (1975) from judgments of computability. With nonsense formulas, much longer times were required to identify noncomputable problems than to compute answers, with a much smaller difference when formulas consisted of meaningful concepts. The better performance on noncomputable problems and questions with meaningful formulas corroborates an interpretation that those items test the connection of algorithms with general conceptual knowledge. Finally, it was found that for relatively complex problems, solution times and time to judge computability were longer if nonsense formulas were learned in separate sets than if they were learned in a single set; however, no such effect was found with meaningful formulas. It was concluded that learning conditions influenced the integration of cognitive structure in the case of nonsense formulas, while subjects were able to adjust organization of the meaningful formulas.     

The neural basis of insight problem solving: an event-related potential study

The electrophysiological correlates of successful insight problem solving (Chinese logogriphs) were studied in 18 healthy subjects using high-density event-related potentials (ERPs). A new experimental paradigm (learning-testing model) was adopted in order to make subjects find a solution on their own initiative rather than receive an answer passively. Results showed that Successful guessed logogriphs elicited a more positive ERP deflection (P200-600) than did Unsuccessful guessed logogriphs in the time window from 200 to 600 ms after onset of the stimuli. Subsequently Successful logogriphs elicited a more negative ERP deflection than did Unsuccessful logogriphs in the time windows of 1500-2000 ms (N1500-2000) and 2000-2500 ms (N2000-2500). Maps of the P200-600 showed strong activity in the midline parieto-occipital scalp regions. Dipole analysis localized the generator of P200-600 in the left superior temporal gyrus and parietotemporo-occipital cortex areas. The N1500-2000 and N2000-2500 had a distinct activation over left frontal scalp regions. Dipole analysis localized the generator of the N1500-2000 in the anterior cingulate cortex (ACC) and the N2000-2500 in the posterior cingulate cortex (PCC). This result indicates that the parietotemporo-occipital cortex areas might be involved in forming rich associations in the early stage of successful logogriph solving. Then, the ACC might play an important role in the breaking mental set and the forming of novel associations. At last, "Aha" feeling might activate the PCC.     

Acquiring an understanding of design: evidence from children's insight problem solving

The human ability to make tools and use them to solve problems may not be zoologically unique, but it is certainly extraordinary. Yet little is known about the conceptual machinery that makes humans so competent at making and using tools. Do adults and children have concepts specialized for understanding human-made artifacts? If so, are these concepts deployed in attempts to solve novel problems? Here we present new data, derived from problem-solving experiments, which support the following. (i) The structure of the child's concept of artifact function changes profoundly between ages 5 and 7. At age 5, the child's conceptual machinery defines the function of an artifact as any goal a user might have; by age 7, its function is defined by the artifact's typical or intended use. (ii) This conceptual shift has a striking effect on problem-solving performance, i.e. the child's concept of artifact function appears to be deployed in problem solving. (iii) This effect on problem solving is not caused by differences in the amount of knowledge that children have about the typical use of a particular tool; it is mediated by the structure of the child's artifact concept (which organizes and deploys the child's knowledge). In two studies, children between 5 and 7 years of age were matched for their knowledge of what a particular artifact "is for", and then given a problem that can only be solved if that tool is used for an atypical purpose. All children performed well in a baseline condition. But when they were primed by a demonstration of the artifact's typical function, 5-year-old children solved the problem much faster than 6-7-year-old children. Because all children knew what the tools were for, differences in knowledge alone cannot explain the results. We argue that the older children were slower to solve the problem when the typical function was primed because (i) their artifact concept plays a role in problem solving, and (ii) intended purpose is central to their concept of artifact function, but not to that of the younger children.     

Goal specificity and knowledge acquisition in statistics problem solving: Evidence for attentional focus

Solving training problems with nonspecific goals (NG; i.e., solving for all possible unknown values) often results in better transfer than solving training problems with standard goals (SG; i.e., solving for one particular unknown value). In this study, we evaluated an attentional focus explanation of the goal specificity effect. According to the attentional focus view, solving NG problems causes attention to be directed to local relations among successive problem states, whereas solving SG problems causes attention to be directed to relations between the various problem states and the goal state. Attention to the former is thought to enhance structural knowledge about the problem domain and thus promote transfer. Results supported this view because structurally different transfer problems were solved faster following NG training than following SG training. Moreover, structural knowledge representations revealed more links depicting local relations following NG training and more links to the training goal following SG training. As predicted, these effects were obtained only by domain novices.     

Mental models and problem solving: Three challenges to a theory of restructuring and insight

The Gestalt psychologists' view of restructuring and the associated phenomenon of insight is discussed and related to findings in modern cognitive psychology. In line with Ohlsson (1984b) it is assumed that search in semantic memory is an indispensable part of restructuring. However, in contrast to Ohlsson's (1984b) information processing theory of restructuring and insight the present paper focuses on the role of mental models. It is asserted that the Gestalt approach to problem solving is compatible with the idea that a mental model is manipulated. The paper discusses three assumptions of restructuring and insight, all of which are related to mental models: (a) restructuring involves manipulating a mental model; (b) the experience of insight is based on "seeing" something in a mental model; (c) restructuring aims at realizing structural balance in a mental model. To assess the validity of these three assumptions is seen as a challenge to future research on human problem solving.     

Intuition versus analysis: Strategy and experience in complex everyday problem solving

Research on dual processes in cognition has shown that explicit, analytical thought is more powerful and less vulnerable to heuristics and biases than is implicit, intuitive thought. However, several studies have shown that holistic, intuitive processes can outperform analysis, documenting the disruptive effects of hypothesis testing, think-aloud protocols, and analytical judgments. To examine the effects of intuitive versus analytical strategy and level of experience on problem solving, 1st- through 4th-year undergraduates solved problems dealing with college life. The results of two studies showed that the appropriateness of strategy depends on the problem solver's level of experience. Analysis was found to be an appropriate strategy for more experienced individuals, whereas novices scored best when they took a holistic, intuitive perspective. Similar effects of strategy were found when strategy instruction was manipulated and when participants were compared on the basis of strategy preference. The implications for research on problem solving, expertise, and dual-process models are discussed.